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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.01%3A_Eigenvalues_and_EigenvectorsThis page explains eigenvalues and eigenvectors in linear algebra, detailing their definitions, significance, and processes for finding them. It discusses how eigenvectors result from matrix transform...This page explains eigenvalues and eigenvectors in linear algebra, detailing their definitions, significance, and processes for finding them. It discusses how eigenvectors result from matrix transformations and the linear independence of distinct eigenvectors. The text covers specific examples, including eigenvalue analysis for specific matrices and the conditions for eigenvalues, including zero.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.07%3A_Basis_and_DimensionThis page discusses the concept of a basis for subspaces in linear algebra, emphasizing the requirements of linear independence and spanning. It covers the basis theorem, providing examples of finding...This page discusses the concept of a basis for subspaces in linear algebra, emphasizing the requirements of linear independence and spanning. It covers the basis theorem, providing examples of finding bases in various dimensions, including specific cases like planes defined by equations. The text explains properties of subspaces such as the column space and null space of matrices, illustrating methods for finding bases and verifying their dimensions.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/MAT_149%3A_Topics_in_Finite_Mathematics_(Holz)/05%3A_Probability/5.04%3A_Markov_Chains/5.4.01%3A_Stochastic_MatricesThis section is devoted to one common kind of application of eigenvalues: to the study of difference equations, in particular to Markov chains. We will introduce stochastic matrices, which encode this...This section is devoted to one common kind of application of eigenvalues: to the study of difference equations, in particular to Markov chains. We will introduce stochastic matrices, which encode this type of difference equation, and will cover in detail the most famous example of a stochastic matrix: the Google Matrix.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/01%3A_Systems_of_Linear_Equations-_AlgebraThis page discusses the algebraic study of linear equations, detailing methods for solving them, particularly through row reduction. It explains a systematic approach to solving equations and how to e...This page discusses the algebraic study of linear equations, detailing methods for solving them, particularly through row reduction. It explains a systematic approach to solving equations and how to express solutions in parametric form. The content is organized into sections that build foundational knowledge on linear equations, algorithms for solutions, and solution representation.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.03%3A_DiagonalizationThis page covers diagonalizability of matrices, explaining that a matrix is diagonalizable if it can be expressed as \(A = CDC^{-1}\) with \(D\) diagonal. It discusses the Diagonalization Theorem, eig...This page covers diagonalizability of matrices, explaining that a matrix is diagonalizable if it can be expressed as \(A = CDC^{-1}\) with \(D\) diagonal. It discusses the Diagonalization Theorem, eigenspaces, eigenvalues, and the significance of linear independence among eigenvectors. Multiple diagonal forms can arise, while geometric and algebraic multiplicities influence diagonalizability.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.02%3A_One-to-one_and_Onto_TransformationsThis page discusses the concepts of one-to-one and onto transformations in linear algebra, focusing on matrix transformations. It defines one-to-one as each output having at most one input and outline...This page discusses the concepts of one-to-one and onto transformations in linear algebra, focusing on matrix transformations. It defines one-to-one as each output having at most one input and outlines examples and theorems related to this property. The text emphasizes that a transformation is onto if every output corresponds to some input.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.03%3A_Linear_TransformationsThis page covers linear transformations and their connections to matrix transformations, defining properties necessary for linearity and providing examples of both linear and non-linear transformation...This page covers linear transformations and their connections to matrix transformations, defining properties necessary for linearity and providing examples of both linear and non-linear transformations. It highlights the importance of the zero vector, standard coordinate vectors, and defines transformations like rotations, dilations, and the identity transformation.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.01%3A_Dot_Products_and_OrthogonalityThis page covers the concepts of dot product, vector length, distance, and orthogonality within vector spaces. It defines the dot product mathematically in \(\mathbb{R}^n\) and explains properties lik...This page covers the concepts of dot product, vector length, distance, and orthogonality within vector spaces. It defines the dot product mathematically in \(\mathbb{R}^n\) and explains properties like commutativity and distributivity. Length is derived from the dot product, and the distance between points is defined as the length of the connecting vector. Unit vectors are introduced, and orthogonality is defined as having a dot product of zero.
- https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/05%3A_Vector_Spaces_and_Subspaces/5.03%3A_Basis_and_DimensionIn order to show that \(\mathcal{B}\) is a basis for \(V\text{,}\) we must prove that \(V = \text{Span}\{v_1,v_2,\ldots,v_m\}.\) If not, then there exists some vector \(v_{m+1}\) in \(V\) that is not ...In order to show that \(\mathcal{B}\) is a basis for \(V\text{,}\) we must prove that \(V = \text{Span}\{v_1,v_2,\ldots,v_m\}.\) If not, then there exists some vector \(v_{m+1}\) in \(V\) that is not contained in \(\text{Span}\{v_1,v_2,\ldots,v_m\}.\) By the increasing span criterion Theorem 2.5.2 in Section 2.5, the set \(\{v_1,v_2,\ldots,v_m,v_{m+1}\}\) is also linearly independent.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/00%3A_Front_Matter
- https://math.libretexts.org/Courses/Irvine_Valley_College/Math_26%3A_Introduction_to_Linear_Algebra/01%3A_Systems_of_Linear_Equations/1.05%3A_Linear_IndependenceRecipe: test if a set of vectors is linearly independent / find an equation of linear dependence. Sometimes the span of a set of vectors is “smaller” than you expect from the number of vectors, as in ...Recipe: test if a set of vectors is linearly independent / find an equation of linear dependence. Sometimes the span of a set of vectors is “smaller” than you expect from the number of vectors, as in the picture below. In the present section, we formalize this idea in the notion of linear independence. Note that in each case, one vector is in the span of the others—so it doesn’t make the span bigger.
